By T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)
The difficulties being solved by means of invariant concept are far-reaching generalizations and extensions of difficulties at the "reduction to canonical shape" of varied is nearly an analogous factor, projective geometry. gadgets of linear algebra or, what Invariant concept has a ISO-year background, which has noticeable alternating classes of progress and stagnation, and alterations within the formula of difficulties, tools of resolution, and fields of program. within the final twenty years invariant thought has skilled a interval of progress, influenced through a prior improvement of the idea of algebraic teams and commutative algebra. it truly is now seen as a department of the speculation of algebraic transformation teams (and below a broader interpretation might be pointed out with this theory). we'll freely use the speculation of algebraic teams, an exposition of which are came upon, for instance, within the first article of the current quantity. we are going to additionally think the reader is aware the elemental suggestions and least difficult theorems of commutative algebra and algebraic geometry; whilst deeper effects are wanted, we'll cite them within the textual content or offer appropriate references.
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Extra info for Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory
Theorem 2 (Haboush). Let ¢J: G -+ GL(V) be a rational representation of the connected reductive group G and let v E V be a non-zero fixed vector. There exists a polynomial function f on V such that f(O) = 0, f(v) = 1 and that f(¢J(g)v) = f(v) for all g E G, v E V. Theorem 1 asserts that in characteristic zero one can take f to be linear. The proof in characteristic p > 0 uses the Steinberg representations (see [J, II. 10]). Example. Assume char(k) = 0 and let G be connected reductive. We have the rational representations A.
The map Uw x B -+ G send- ing (u, b) to uwb is an isomorphism of varieties; (iii) The closure C(w) is a union of certain C(w'). 2. The proof of the theorem gives the following result. A. Springer 34 Corollary. For s E S, WE W we have C(s)C(w) = C(sw) if l(sw) > C(s)C(w) = C(sw) u C(w) l(w), if l(sw) < l(w). This implies that (G, B, N, S) is a Tits system in the sense of [B02, Ch. IV]. It follows from the theorem that dim C(w) = l(w) + dim B. It also follows that there is one open C(w), namely C(wo).
Let X be an F-variety. Then X(Fs) is dense in X. 3]. 3. Criteria for Fields of Definition. X is an F -variety. The Galois group = Gal(Fs/F) acts on X(Fs). Proposition. Let Y be a closed subvariety of X. 4]. Corollary 1. The irreducible components of X are defined over Fs. Corollary 2. Let qJ: X -+ Y be an F-morphism. Then qJ(X) is defined over F. A. 4. Proposition. Let Y and Z be closed F-subvarieties of X. The intersection Y n Z is defined over F if one of the following conditions holds: (a) F is perfect, (b) there is a dense subset U of Y n Z such that for all x E U we have TAY n Z) = Tx Y n TxZ.
Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory by T. A. Springer (auth.), A. N. Parshin, I. R. Shafarevich (eds.)